Prediction of crossing nodal-lines and large intrinsic spin Hall conductivity in topological Dirac semimetal Ta3As family

Topological insulators (TIs) are considered as ideal platforms for generating large spin Hall conductivity (SHC), however, the bulk carrier problem, which is unavoidable in TIs, hinders their practical applications. Recently, topological semimetals (TSMs) have been proposed to achieve large SHC to replace TIs. However, the ideal TSM candidates with large SHC are still lacking. In terms of first-principles calculations, we predict that Ta3As family compounds exhibit complex crossing nodal-lines (CNL) properties in absence of the spin-orbit coupling (SOC). However, they transfer to Dirac TSMs under the influence of strong SOC, and present large SHC around Fermi level in particular. Remarkably, the SHC value of Ta3Y (Y = As, Sb, Bi) is around 1500–1700 (ħ/e)(Ω⋅cm)−1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\hbar /e)({\mathrm{{\Omega}}} \cdot {\mathrm{cm}})^{ - 1}$$\end{document}, which is comparable to noble metal Pt and much larger than TIs, Weyl TSMs, and 4d/5d transition metals. Our work not only suggests a kind of TSM family with interesting Dirac CNL around Fermi level, but also paves the way for searching large intrinsic SHC materials in complex CNL TSM systems.


INTRODUCTION
The spin Hall effect (SHE) is a phenomenon that a transverse pure spin current appears if a longitudinal electric field is applied [1][2][3][4] . Topological insulators (TIs) and quantum SHE systems were considered as ideal materials for producing pure spin current 5,6 . However, the experiments have shown that the bulk carrier problem prevents the widespread applications of TIs in spintronic devices because of the hybridization between surface states and bulk states in Bi 2 Se 3 class materials 7,8 . Generally speaking, there are two kinds of mechanisms for the SHE, which are the extrinsic mechanism and intrinsic mechanism. For the former, impurities and defects with strong SOC are the main scattering resources. Both the side-jump and skew scattering are specifically identified and similar to those in the study of anomalous Hall effect. The intrinsic mechanism roots in the topologically nontrivial energy bands, which can be accurately calculated based on ab initio calculations and Kubo formula. Recently, heavy metals and compounds including heavy elements have attracted growing attention for the possibility to realize large SHE [9][10][11][12][13] . In particular, a lot of interests focus on topological semimetals (TSMs), which exhibit similar spin-momentum locking in both bulk and topological surface states. To date, the SHC of some TSMs, like TaAs, NbAs, and rutile oxides 9,14 , is still far below that of pure heavy metal Pt and W. Therefore, the TSMs with larger SHC are still highly desirable both for the purposes of fundamental research and technical applications.
According to the distribution and degeneracy of crossing nodal points, TSMs can usually be divided into three categories: Dirac semimetals, Weyl semimetals, and nodal line semimetals (NLSMs) [15][16][17][18][19][20][21][22][23] . The Weyl points are discrete points in momentum space where the conduction bands and valence bands cross each other. However, in the NLSMs, these crossing points are not discrete but form a continuous path. Compared to the Fermi arc surface states existing in Dirac and Weyl semimetals, NLSMs show drumhead surface states, which are regarded as excellent platforms hosting many interaction-induced nontrivial states, such as superconductivity and fractional topological insulator 21 . It is known that the nodal line can be protected by appropriate symmetries, such as mirror symmetry, time-reversal symmetry, and inversion symmetry [24][25][26] . Interestingly, the nodal line states have been classified into various types, e.g., isolated closed nodal ring, nodal chain, nodal link, nodal knots, nodal straight line, and even crossing nodal-line (CNL) 27,28 . Until now, the real NLSMs that can host CNL are very rare 28 . It is curious to find these CNL semimetals (CNLSMs) and explore their possibility on exotic SHC applications.
In this article, using first-principles calculations, we propose that the X 3 Y (X = Ta, Nb; Y = As, Sb, Bi) compounds are a family of CNLSMs. Interestingly, without taking into account the effects of SOC, X 3 Y compounds host two sets of complex CNL around Fermi level. One is located around the corners, while others are distributed in the bulk of momentum space. When the SOC is present, Dirac points are emergent along the R-M path due to the C 4 rotational symmetry and are absent along other directions with a very tiny gap. Importantly, it is found that Ta 3 Y compounds exhibit large intrinsic SHC. For example, the SHC is around 1500 ð h=eÞ Ω Á cm ð Þ À1 in stable Ta 3 As compound, which is attributed to its large SOC band splitting. Therefore, the prediction of Dirac CNLSMs in X 3 Y family not only extends our current knowledge on NLSMs, but also suggests a routine of realizing large SHC to replace noble metals. Significantly, X 3 Y family materials, in particular the stable Ta 3 As compound, are promising candidates for the application in spintronic devices [29][30][31][32] .

RESULTS AND DISCUSSION Crystal structures and stability of X 3 Y compounds
The crystal structure of the X 3 Y (X = Ta, Nb; Y = As, Sb, Bi) compounds is the same as the ordered L12 phase Cu 3 Au 16,33 . We adopt the cubic structure with the space group Pm3m (No. 221), as shown in Fig. 1a. A primitive cell contains three X atoms at the plane center and one Y atom at the vertex. Figure 1b shows the bulk Brillouin zone (BZ) of the X 3 Y and the projected surface BZ of the (010) surface. Previous studies mainly focused on the Ta 3 Y compounds with the Ti 3 P-type (A15 phase, space group 223) structure 34,35 , instead of the Pm3m phase. To illustrate the thermodynamic and lattice stability of Pm3m-phase X 3 Y compounds, we calculate their formation enthalpies (ΔH) and the phonon spectra. The negative ΔH values of Ta 3 As, Nb 3 As, and Nb 3 Sb indicate that they are thermodynamically stable against phase decomposition, as shown in Fig. 1c. In addition, the absence of imaginary modes in the entire BZ further confirms their lattice stability, as shown in Supplementary Fig. 1. The calculated lattice constant of Ta 3 As, Nb 3 As, and Nb 3 Sb are 4.102, 4.105, and 4.226 Å, respectively.

Electronic structures and effective Hamiltonian model
The calculated electronic structures of Ta 3 As, Nb 3 As, and Nb 3 Sb present similar characters, with fourfold degeneracy around a crossing point because of the time-reversal and inversion symmetry. Without the SOC effects, Dirac points exist along high-symmetry paths in the BZ, e.g., M-X and M-Γ paths, as shown in Fig. 2. Similar to the case of Cu 3 PdN and Mackay-Terrones crystal, the band crossings can be understood from the argument of codimension [16][17][18][19][20][21] . In general, the four bands without SOC effects near the Dirac points can be described by two identical 2 × 2 Hamiltonians, which are written as Where σ x,y,z and d x,y,z (k) are Pauli matrices and corresponding coefficients, respectively. At Γ and R point with little group O h , the time-reversal, inversion symmetry as well as rotation symmetry restrict the coefficients, with d y (k) = 0, d x (k), and d z (k) being odd and even functions correspondingly. We can write d x,z (k) as follow, Here γ, λ x,y,z , M, and B are the corresponding coefficients. The p . The band crossing points appear only for the case of E(k) = 0, i.e., both d x (k) = 0 and d z (k) = 0. Here, the condition d z (k) = 0 is satisfied only when energy bands inversion occurs. As shown in Supplementary Fig. 2, it is clear that the p-d bands inversion happens along the R-Γ, R-X, and R-M paths. Furthermore, if we ignore the higher-order terms in Eq. (2) with d x (k) = γk x k y k z = 0, Eq.
(2) and Eq. (3) induce a nodal line topological state 16 . However, the crossing points are gapped along M-X, M-Γ, R-Γ, and R-X paths but preserved along R-M path when taking into account the SOC in Fig. 2. The band structures are similar for Ta 3 As, Nb 3 As, and Nb 3 Sb. Clearly, both the R and Γ points belong to the O h double point group, and the irreducible representation of bands near the Dirac points is marked with the red symbol Γ i (i = 4, 5, 6, 7). (i) The R-X path: Moving from R to X, the symmetry is lowered to C 2v double point group, where only the Γ 5 irreducible representation exists 36 . (ii) The R-Γ path: Characterized by C 3v double group, both the occupied band and non-occupied one along R-Γ path are the Γ 4 irreducible representation in C 3v double point group. As we know, if the two bands at the crossing points belong to the same irreducible representation, the crossing points will not be protected by the symmetry 37 . In Fig. 2a (without SOC) and Fig. 2d (with SOC), it is observed that the crossing points are gapped with a tiny value along R-X and R-Γ path due to the effect of SOC. (iii) The R-M path: The crossing points still exist, which results from the non-occupied and occupied bands belonging to different irreducible representations Γ 6 and Γ 7 in the C 4v double point group, respectively. The existing crossing points are protected by crystal symmetry, thus no gap appears along the R-M path. To express the influence of SOC strength on Dirac points, we take Ta 3 As as an example to show the evolution of energy bands at different SOC strengths along the R-X and R-M path in Supplementary Fig. 4. When the SOC strength is artificially increased to three times larger than the intrinsic value, the Dirac points disappear obviously along the R-Γ path but still exist along the R-M path.

Crossing nodal-lines (CNL) and drumhead surface states
In previous research for Dirac nodal line and nodal sphere, the energy gap is E g < 2 meV 28 . This energy scale is rough and it is desirable to explore the behavior of Dirac nodal points more precisely. Therefore, we explore extremely tiny energy gaps which are identified at 0.1 meV level, and present their distributions in 3D momentum space as well as the topological surface states in Fig. 3. It is clearly seen that continuous Dirac nodal points are merging into nodal lines, and they cross each other forming the crossing nodal-lines. Moreover, three nodal lines intersect at the R point, while two lines form a nodal net centered on the Γ point. And because of the same O h double group at both points, the crossing nodal-lines appear at the position of the R point and Γ point. As we know, the NLSMs should present drumhead surface states, which can be observed in Fig. 3b with the connection between conduction and valence bands on the (010) surface 21 . Besides, the surface states on other oriented surfaces, like (100) and (001) surface, are also calculated, as shown in Supplementary  Fig. 7. It is observed that the drumhead surface states on the above three surfaces are almost the same, which is similar to the case in YH 3 materials 28 .

Spin Hall conductivity
As a measurable and calculated physical quantity, the intrinsic spin Hall conductivity (SHC) is usually used to describe the strength of the SHE. The SHC and SHE in TSMs have been explored in a large number of previous works. It is indicated that band crossings may lead to large SHC 9,14 . In view of band characters, we also find and analyze large SHC around the Fermi energy in the Ta 3 Y (Y = As, Sb, Bi) compounds. At the zero-temperature limit, the Fig. 1 Crystal structure. a The atomic structure of the L12 phase X 3 Y (X = Ta, Nb; Y = As, Sb, Bi) family compounds. Blue and yellow spheres denote the X and Y atoms, respectively. b The bulk Brillouin zone (BZ) and the projected BZ on the (010) surface. c The formation enthalpies (ΔH) of different X 3 Y compounds. The unit is eV per unitcell.
intrinsic SHC tensor can be expressed using the Kubo formula 38,39 , Here,ĵ x ¼ 1 2 ½ŝ z ;v x is the spin current operator, V is the primitive cell volume, N 3 k is the number of k points in the BZ, n (m), ϵ n and | mk〉 are band indexes, eigenvalues and the Bloch wave function, respectively, f nk is the Fermi distribution function, andv y ¼ 1 h ∂HðkÞ ∂ky is the velocity operator. Besides, we multiply a factor 2e= h to the SHC in the unit of charge conductivity (Ω·cm) −1 , similar to anomalous Hall conductivity 40 .
Further, we analyze the mechanism by calculating the bandresolved and momentum-resolved SHC of the stable Ta 3 As compound, corresponding to Fig. 4b, c. Here, the contribution to SHC is noted with red color dots in Fig. 4b, and the momentumresolved SHC after taking the logarithm is shown in Fig. 4c. From Fig. 4 and Fig. 2 as well as Supplementary Fig. 5, we observe that the states around Δ point (along X-M path) mostly contribute to the large SHC with strong SOC effect. Note that the SHC of CNLSMs Ta 3 Y compounds is much larger than that of Nb 3 Y. Basically, the element Ta is heavier than Nb. Thus, the SOC in Ta 3 Y compounds may be stronger than those in Nb 3 Y, leading to larger SHC in CNLSMs Ta 3 As family. Moreover, the SHC of Ta 3 Bi compound almost arrives to 1700 ð h=eÞðΩ Á cmÞ À1 and becomes the largest one in the Dirac CNLSMs X 3 Y compounds, thanks to the larger atomic mass of element Bi than other members in the same group. In brief, the stronger SHE and larger SHC in X 3 Y family   The band-resolved SHC of Ta 3 As. c The momentum-resolved SHC of Ta 3 As along the highsymmetry momentum paths. materials stem from larger atomic mass and stronger atomic SOC in these compounds. In addition, we show the same character of band structures and the effect of SOC on band inversion in different X 3 Y materials in Supplementary Fig. 8.
In summary, based on the first-principles calculations, we have predicted the X 3 Y (X = Ta, Nb; Y = As, Sb, Bi) family compounds as a kind of Dirac crossing nodal-line semimetal (CNLSMs) with large spin Hall conductivity (SHC), which can serve as an ideal platform to research rich topological quantum states and explore the application in spintronics. In the absence of SOC, three nodal lines converge at the R point and two lines cross with forming a nodal net around the Γ point. After SOC is included, Dirac points are protected along the R-M path by the C 4 rotational symmetry and open a very tiny gap along other directions. The drumhead surface states conform to the feature of Dirac CNLSMs. At the same time, by calculating the momentum-resolved and bandresolved spin Hall conductivity, the large SHC in Ta 3 As is attributed to the SOC along X-M path, instead of the nodal points. This mechanism differs from both the Weyl point leading to SHC for TaAs and nodal line inducing SHC for IrO 2 9,14 . The prediction in this work should promote further research on spin Hall effect, the topological and superconductivity properties, as well as quantum phase transitions. On the other hand, the fascinating crossing nodal-lines and the remarkable SHC at Fermi energy will contribute to their practical applications in assembling spintronic devices such as magnetic spin-orbit torque random access memory.

METHODS DFT calculations
The first-principles calculations are performed within the framework of density functional theory (DFT), as implemented in QUANTUM ESPRESSO package 42 . The generalized gradient approximation in the parametrization of Perdew-Burke-Ernzerhof is used for the exchange-correlation potential 43 . The kinetic energy cut-off and the charge density cut-off of the plane wave basis are chosen with 70 Ry and 850 Ry, respectively. All atoms are relaxed until the forces are smaller than 0.001 eV/Å. An 41 × 41 × 41 k-point mesh in the electronic self-consistent calculations is used for the Brillouin zone (BZ) integration. To evaluate the thermodynamic stabilities of X 3 Y compounds, the formation enthalpies (ΔH) were calculated as: ΔH = E (X 3 Y)-3E(X)-E(Y). The phonon spectra were calculated using the PHONOPY codes based on the finite displacement method within a 4 × 4 × 4 supercell with 3 × 3 × 3 k-point mesh 44 . The spin Berry curvature, spin Hall conductivity, and topological edge states are calculated by using a tightbinding Hamiltonian constructed on a basis of maximally localized Wannier functions implemented in Wannier90 and WannierTools codes 45,46 . The SHC is obtained by summing the spin Berry curvatures over all the occupied bands in a dense k-point grid of 200 × 200 × 200.

DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.